Method for generating a signal by means of a turbo-encoder, and corresponding device and computer program

ABSTRACT

A method for generating a signal, including turbo-coding a set of information symbols delivering, on the one hand, the information symbols and, on the other hand, redundancy symbols. The turbo-coding implementing, to obtain the redundancy symbols: an encoding of the set of information symbols by a first encoder, an interleaving of the set of information symbols, and an encoding of the set of information symbols interleaved by a second encoder. The turbo-coding also implements a bijective transformation of the information symbols, implemented before and/or after the interleaving, the transformation modifying a value of at least two of the information symbols prior to the coding of the information symbols by the first and/or the second coder.

CROSS-REFERENCE TO RELATED APPLICATIONS

This Application is a Section 371 National Stage Application ofInternational Application No. PCT/FR2019/053048, filed Dec. 12, 2019,which is incorporated by reference in its entirety and published as WO2020/128248 A1 on Jun. 25, 2020, not in English.

1. FIELD OF THE INVENTION

The field of the invention is that of digital communications,implementing a turbo-code type error correcting code.

In particular, the invention provides a new technique for generating asignal implementing a non-binary turbo-code type error correcting code.

The invention finds applications in particular in the field of wireless(DAB, DVB-T, WLAN, unguided optics, etc.) or wired (xDSL, PLC, optics,etc.) communications.

2. PRIOR ART

2.1 Binary Turbo-Coders

The error correcting codes conventionally used in digital communicationsystems are binary. In other words, the information symbols at the inputof the encoder, and the redundancy symbols obtained at the output of theencoder, are binary elements, belonging to the Galois field GF(2).

The conventional structure of a binary turbo-coder, illustrated in FIG.1 , comprises the parallel concatenation of two encoders, for example ofthe recursive systematic convolutional type, C1 11 and C2 12, separatedby an interleaver 13. The input digital message D, formed of Kinformation symbols defined on GF(2), is encoded in the order of arrivalof the information symbols, called natural order, by the first encoderC1 11, and in an interleaved or permuted order, by the second encoder C212. The digital output message comprises systematic symbols S,corresponding to the K information symbols, and parity or redundancysymbols P1 and P2, defined on GF(2).

2.2 Non-Binary Turbo-Coders

Error correcting codes can also be defined on Galois fields withcardinal q greater than 2, GF(q). The structure of a non-binaryturbo-coder is similar to the structure of a binary turbo-coder. In thiscase, the digital input message D is formed of K information symbolsdefined on GF(q), and the output digital message is formed of systematicsymbols S, corresponding to the K information symbols, and of parity orredundancy symbols P1 and P2, defined on GF(q).

These non-binary error correcting codes can be associated with amodulation of order p equal to q (for example, quadrature amplitudemodulation q-QAM) or with a modulation of different order. Inparticular, the correction power of the non-binary codes is greater thanthe correction power of the binary codes when these codes are associatedwith a modulation of the same order, p equal to q (for example q-QAM).

A difference between binary and non-binary codes from the point of viewof corrective power is that the low error rate performances of a binarycode are dictated by its minimum Hamming distance, while the low errorrate performances of a non-binary code are dictated by the minimumcumulative Euclidean distance between two coded sequences (that is tosay two turbo-coder output messages), which takes into account theconstellation of the p-ary modulation used.

When combined with a modulation of the same order p=q (for exampleq-QAM), the high signal-to-noise ratio correction power on a Gaussianadditive noise channel of a non-binary convolutional code defined onGF(q), in terms of Frame Error Rate (FER), is given by the Union Bound(UB):

$\begin{matrix}{{{TEP} \leq {UB}} = {\frac{1}{C_{q^{K}}^{2}}{\sum\limits_{d = d_{\min}}^{d_{\max}}{{n(d)}{Q\left( \frac{d}{2\sigma} \right)}}}}} & \left\lbrack {{Equation}1} \right\rbrack\end{matrix}$

where K is the length of the digital input messages,

C_(q) _(K) ² is the number of combinations of pairs of digital inputmessages of length K,

d_(min) is the minimum cumulative Euclidean distance between two outputmessages (or code words),

n(d) is the multiplicity associated with the distance d, that is to saythe number of pairs of code words at the distance d from each other,

σ is the variance of the Gaussian noise, and

Q(⋅) denotes the error function:

${Q(x)} = {\frac{1}{\sqrt{2\pi}}{\int_{x}^{\infty}{e^{- t^{2/2}}d{t.}}}}$

The value of the bound UB is dominated by its first terms, correspondingto the low values of the distance d, namely d_(min) and the values justabove.

3. DESCRIPTION OF THE INVENTION

In order to improve the performances of turbo-codes, in particularnon-binary turbo-codes, it is desirable to increase the distance betweenthe output messages. For example, it is desirable to increase theminimum cumulative Euclidean distance between two code words compared tothe prior art.

For this purpose, the invention provides a method for generating asignal, comprising a step of turbo-coding a set of information symbolsdelivering, on the one hand, the information symbols and, on the otherhand, redundancy symbols, implementing, to obtain the redundancysymbols:

-   -   an encoding of said set of information symbols by a first        encoder, delivering a first set of redundancy symbols,    -   an interleaving of said set of information symbols, delivering a        set of interleaved information symbols,    -   an encoding of said set of interleaved information symbols by a        second encoder, delivering a second set of redundancy symbols.

According to the invention, the turbo-coding step also implements abijective transformation of said information symbols, implemented beforeand/or after said interleaving, said transformation modifying the valueof at least two of said information symbols prior to the encoding ofsaid information symbols by the first and/or the second encoder.

In other words, the invention proposes a modification of theconventional structure of turbo-coders, based on the addition of afunction for transforming the information symbols before the encoding ofthe input message by the first encoder or by the second encoder, andimplemented either before or after interleaving.

In this way, the order of the information symbols entering on the twoencoders is changed on the one hand, thanks to the interleaving, and onthe other hand the value of the information symbols entering the twoencoders is changed, thanks to the transformation. The coded sequencesobtained at the output of the two encoders therefore do not consist ofthe same information and redundancy symbols.

The transformation considered is a bijection on the considered set ofsymbols of the Galois field: each symbol of GF(q) has a unique image inGF(q) by this transformation and every element of GF(q) has a uniqueantecedent in GF(q). Such a transformation therefore modifies the valueof at least two information symbols.

For example, if the information symbols belong to the Galois field GF(2)and if a set of information symbols forming the input message 000000100is considered, the set of interleaved information symbols can be010000000 before transformation, and 101111111 after transformation (thebijective transformation transforming, in this case, information symbolsof value “0” into information symbols of value “1” and vice versa). Itis therefore possible to change both the order and the value of theinformation symbols input on the two encoders of the turbo-coder.

The overall objective of this transformation is to increase the minimumcumulative distance between two code words, for example the cumulativeEuclidean distance, relative to the turbo-coder without transformation,independently of the nature of the elementary codes implemented by thefirst and second encoders and interleaver used.

A solution is thus proposed which allows, according to at least oneembodiment, to improve the corrective power of the turbo-decoder.

In particular, the information symbols belong to a Galois field ofcardinal q, denoted by GF(q), with q>2. Indeed, the implementation of atransformation function allows in particular to improve the correctionperformances of non-binary turbo-codes.

According to a particular embodiment, the transformation generates aminimum dispersion Δ_(min) between two information symbols S_(i), S_(j)among said information symbols greater than a selection threshold, suchthat:Δ_(min)=min Δ(S _(i) ,S _(j))with:

-   -   Δ(S_(i),S_(j))=D(S_(i),S_(j))+D(T(S_(i)),T(S_(j)))    -   D(S_(i),S_(j)) the distance between said information symbols        S_(i), S_(j) before transformation,    -   D(T(S_(i)),T(S_(j))) the distance between said information        symbols S_(i), S_(j) after transformation by the function T.

In other words, the proposed solution seeks to maximise the cumulativegap between information symbols before transformation and informationsymbols after transformation.

According to a particular embodiment, the method implements a step ofmapping the information and redundancy symbols onto modulation symbolsassociated with a constellation of order p, an information or redundancysymbol being mapped onto at least one modulation symbol, and thetransformation takes into account said constellation.

In particular, the transformation takes into account the position of themodulation symbols associated with the information symbols before andafter transformation, and seeks to increase the Euclidean distancebetween these modulation symbols compared to an implementation withouttransformation.

According to a first exemplary embodiment, the order p of theconstellation being equal to the cardinal q of the Galois field to whichthe information symbols belong, each information or redundancy symbol ismapped onto a single modulation symbol.

According to this first example, the dispersion between two informationsymbols S_(i), S_(j) is expressed in the form:Δ(S _(i) ,S _(j))=d _(euc) ²(S _(i) ^(m) ,S _(j) ^(m))+d _(euc) ²(T(S_(i))^(m) ,T(S _(j))^(m))with:

-   -   d_(euc) ²(S_(i) ^(m),S_(j) ^(m))=(I_(s) _(i) _(m) −I_(s) _(j)        _(m) )²+(Q_(s) _(i) _(m) −Q_(s) _(j) _(m) )² the square of the        Euclidean distance between the modulation symbols S_(i) ^(m),        S_(j) ^(m) onto which said information symbols S_(i), S_(j) are        mapped before transformation;    -   d_(euc) ²(T(S_(i))^(m),T(S_(j))^(m))=(I_(T(S) _(i) ₎ _(m)        −I_(T(S) _(j) ₎ _(m) )²+(Q_(T(S) _(i) ₎ _(m) −Q_(T(S) _(j) ₎        _(m) )² the square of the Euclidean distance between the        modulation symbols T(S_(i))^(m), T(S_(j))^(m) onto which are        mapped said information symbols S_(i), S_(j) after        transformation; and    -   I_(x) and Q_(x) are the in-phase and quadrature components of a        signal x in the considered constellation.

In other words, the distance D(S_(i), S_(j)) between the informationsymbols S_(i), S_(j) before transformation, and the distance D(T(S_(j)),T(S_(j))) between the information symbols T(S_(j)), T(Sj) aftertransformation, are expressed in the form of squared Euclidean distance.

According to a first criterion for constructing the transformationfunction, the transformation transforms a pair of information symbolsintended to be mapped onto a pair of modulation symbols whose Euclideandistance is less than a first threshold, into a pair of informationsymbols intended to be mapped onto a pair of modulation symbols whoseEuclidean distance is greater than the first threshold, and vice versa.

According to a second criterion for constructing the transformationfunction, which can be taken individually or in combination with thefirst criterion, the transformation transforms an information symbolintended to be mapped onto a modulation symbol having, in theconstellation, a number of neighbouring modulation symbols less than adetermined number, into an information symbol intended to be mapped ontoa modulation symbol having, in the constellation, a number of neighboursgreater than said determined number, and vice versa.

The method can also implement a step of selecting a transformation fromseveral available transformations, said selection taking into accountsaid redundancy symbols and/or said interleaver.

In this case, the selection can involve properties of the trellisrepresentative of the turbo-code.

According to a second exemplary embodiment, the order p of theconstellation being less than the cardinal q of the Galois field towhich said information symbols belong, each information or redundancysymbol is mapped onto n modulation symbols, with n≥2.

According to this second example, the dispersion between two informationsymbols S_(i), S_(j) is expressed in the form:

${D\left( {S_{i},S_{j}} \right)} = {{{\sum}_{k = 1}^{n}{d_{euc}^{2}\left( {S_{i,k}^{m},\ S_{j,k}^{m}} \right)}} = {{\sum}_{k = 1}^{n}\left\lbrack {\left( {I_{S_{i,k}^{m}} - I_{s_{j,k}^{m}}} \right)^{2} + \left( {Q_{S_{i,k}^{m}} - Q_{S_{j,k}^{m}}} \right)^{2}} \right\rbrack}}$$\begin{matrix}{{D\left( {{T\left( S_{i} \right)},{T\left( S_{j} \right)}} \right)} = {\sum\limits_{k = 1}^{n}{d_{euc}^{2}\left( {{T\left( S_{i} \right)}_{k}^{m},{T\left( S_{j} \right)}_{k}^{m}} \right)}}} \\{= {\overset{n}{\sum\limits_{k = 1}}\left\lbrack {\left( {I_{{T(S_{i})}_{k}^{m}} - I_{{T(S_{j})}_{k}^{m}}} \right)^{2} + \left( {Q_{{T(S_{i})}_{k}^{m}} - Q_{{T(S_{j})}_{k}^{m}}} \right)^{2}} \right\rbrack}}\end{matrix}$

-   -   S^(m) _(i,k) respectively S^(m) _(j,k), for k ranging from 1 to        n, the n modulation symbols associated with the information        symbol S_(i), respectively S_(j), before transformation,        T(S_(i))_(k) ^(m), respectively T(S_(j))_(k) ^(m), for k ranging        from 1 to n, the n modulation symbols associated with the        information symbol S_(i), respectively S_(j), after        transformation, I_(x) and Q_(x) are the in-phase and quadrature        components of a signal x in the considered constellation.

According to a first criterion for the construction of thetransformation function, the transformation minimises the number of zeroterms in the expression of the dispersion Δ(S_(i),S_(j)).

The method can also implement a step of selecting a transformation fromseveral available transformations, said selection maximising the valueof the non-zero terms in the expression of the dispersionΔ(S_(i),S_(j)).

According to one embodiment, the first encoder implements at least afirst elementary code and the second encoder implements at least asecond elementary code, distinct from said first elementary code. Theturbo-coder can thus use elementary codes having different correctingpower.

According to another embodiment, the first and second elementary codesare identical.

The invention also relates to a corresponding device for generating asignal, comprising a turbo-encoder configured for encoding a set ofinformation symbols and delivering, on the one hand, said informationsymbols and, on the other hand, redundancy symbols, said turbo-encodercomprising:

-   -   a first encoder for coding said set of information symbols,        delivering a first set of redundancy symbols,    -   an interleaver for interleaving said set of information symbols,        delivering a set of interleaved information symbols,    -   a second encoder for coding said set of interleaved information        symbols, delivering a second set of redundancy symbols,

The turbo-encoder also comprises a processor for applying a bijectivetransformation to said information symbols, implemented before and/orafter said interleaver, said transformation modifying the value of atleast two of said information symbols prior to the encoding of saidinformation symbols by the first and/or the second encoder.

Such a device for generating a signal is in particular adapted toimplement the generation method described above. This is, for example, abinary or non-binary turbo-coder. This device could of course includethe various features relating to the generation method according to theinvention, which can be combined or taken in isolation. Thus, thefeatures and advantages of this device are the same as those of themethod described above. Therefore, they are not further detailed.

The invention also relates to one or more computer programs includinginstructions for the implementation of a method for generating a signalas described above when this or these programs are executed by at leastone processor.

The invention also relates to an information medium readable by acomputer, and including instructions of a computer program as mentionedabove.

4. LIST OF FIGURES

Other features and advantages of the invention will emerge more clearlyupon reading the following description of a particular embodiment, givenby way of simple illustrative and non-limiting example, and the appendeddrawings, among which:

FIG. 1 illustrates a structure of a turbo-coder according to the priorart.

FIGS. 2A and 2B illustrate two examples of a turbo-coder according tovarious embodiments of the invention.

FIGS. 3A and 3B respectively illustrate the generic structure of aconvolutional coder with a memory element on GF(q), and an example of atrellis of such a code.

FIG. 4 shows some DC sequences of a code trellis.

FIG. 5 illustrates a constellation associated with a 64-state amplitudemodulation.

FIG. 6 illustrates the points in the constellation of FIG. 5 thatexperience the least interference.

FIGS. 7A to 7C compare the correction performances of a turbo-codeaccording to one embodiment of the invention with a turbo-code accordingto the prior art, when the order of the modulation and of the turbo-codeare the same.

FIG. 8 compares the correction performances of a turbo-code according toone embodiment of the invention with a turbo-code according to the priorart, when the order of the modulation and of the turbo-code aredifferent.

FIG. 9 shows the simplified structure of a device implementing atechnique for generating a signal according to one embodiment of theinvention.

5. DESCRIPTION OF AN EMBODIMENT OF THE INVENTION

5.1 General Principle

The general principle of the invention is based on the addition of afunction for transforming information symbols in a turbo-coder, inparticular a “non-binary” turbo-coder, allowing to modify the value ofat least two information symbols, so that the sequence input on oneencoder of the turbo-coder is different from the sequence input on theother encoder.

Two examples of structures of a turbo-coder implementing theturbo-coding step according to the invention are illustrated in FIGS. 2Aand 2B. These two examples implement a first encoder C1 21 and a secondencoder C2 22 in parallel, for example of the recursive systematicconvolutional type, separated by an interleaver 23 and a transformationmodule 24. The digital input message D comprises K information symbols,and the digital output message comprises systematic symbols S and parityor redundancy symbols P1 and P2. Each sequence composed of the Kinformation symbols and the P1 and P2 redundancy symbols forms a codeword.

The systematic symbols S correspond to the K information symbols,possibly after interleaving and/or transformation.

According to the example illustrated in FIG. 2A, the digital inputmessage D is encoded by the first encoder C1 21A, in the order ofarrival of the information symbols. The input digital message D is, inparallel, transformed in the transformation module 24A, and thetransformed information symbols are interleaved by the interleaver 23A.The interleaved transformed information symbols are then encoded by thesecond encoder C2 22A.

According to the example illustrated in FIG. 2B, the digital inputmessage D is transformed in the transformation module 24B, and thetransformed information symbols are encoded by the first encoder C1 21B,in the natural order. The input digital message D is, in parallel,interleaved by the interleaver 23B, and then the interleaved informationsymbols are encoded by the second encoder C2 22B.

Note that according to another example, not illustrated, thetransformation can be implemented after the interleaving. Thetransformation can therefore be implemented either before or after theinterleaving.

According to yet another example, not illustrated, a firsttransformation can be implemented before the first encoder, before theinterleaver, and a second transformation, different from the firsttransformation and not cancelling the first transformation, can beimplemented before the second encoder, before or after the interleaver.In this case, it is considered that the transformation function has twocomponents, a first component corresponding to the first transformationand a second component corresponding to the second transformation. The“overall” transformation is therefore composed of a combination of thefirst and second transformations.

The term “transformation” is used below to cover these variousimplementations.

The turbo-coding step according to the invention therefore implements abijective transformation of the information symbols, implemented beforeand/or after the interleaving, said transformation modifying the valueof at least two of the information symbols prior to the encoding of theinformation symbols by the first and/or the second encoder.

In this way, the sequence of information symbols entering on the firstencoder C1 differs from the sequence of information symbols entering onthe second encoder C2, on the one hand by the order of the informationsymbols entering on each encoder, thanks to the interleaving, on theother hand by the value of the information symbols entering on eachencoder, thanks to the transformation.

As indicated previously, the considered transformation is a bijection onthe set of symbols of the considered Galois field. For example, if theinformation symbols are considered to be able to take values rangingfrom 0 to 15, that is to say are defined on a Galois field of cardinalq=16, then each information symbol of GF(16) has a unique image inGF(16) by the transformation T, and every element of GF(16) has a uniqueantecedent in GF(16). The table below gives an example of the bijectivetransformation of information symbols S_(i), defined in GF(16), intoinformation symbols T(S_(j)), defined in GF(16):

TABLE 1 S_(i) T(S_(i)) 0 9 1 5 2 7 3 4 4 12 5 11 6 15 7 1 8 13 9 3 10 1411 10 12 2 13 0 14 8 15 6

In particular, if information symbols belonging to a Galois field ofcardinal q with q>2, denoted GF(q), are considered, the proposedturbo-coding step allows to increase the distance between theinformation symbols, and, consequently, the minimum distance between thecode words output by the turbo-coder.

According to a particular embodiment, the overall objective of thetransformation is to increase the minimum cumulative Euclidean distancebetween two sequences of the resulting code relative to the code withouttransformation, independently of the nature of the elementary codesimplemented by the first and second encoders and interleaver used.

Note that the minimum cumulative Euclidean distance between two codewords output by the turbo-coder depends in particular on theconstellation of the modulation used to transmit the code words.

Thus, according to one embodiment, the generation method according tothe invention comprises a step of mapping information and redundancysymbols onto modulation symbols associated with a constellation of orderp.

The transformation can thus be selected taking into account theconstellation, and in particular the position of the modulation symbolsassociated with the information symbols before and after transformation.

5.2 Identification of “Problematic” Information Symbols

Conventionally, non-binary turbo-coders use recursive systematicconvolutional encoders implementing elementary codes with a singlememory element (v=1). Indeed, the complexity of decoding a code on GF(q)varies in q^(v), where v is the memory of the code, and it is thereforeeasier to decode a code having only one memory element.

These convolutional encoders have the particularity of having fullyconnected trellises, that is to say that all the states are connected toeach other in the code trellis.

For example, FIG. 3A illustrates the generic structure of aconvolutional coder with a memory element on GF(q), with a₁, a₂, a₃ codesetting parameters, D one memory element, I the input informationsymbols, S the output systematic symbols and P the output paritysymbols. FIG. 3B illustrates an example of a trellis for a code definedon GF(4), with particular values for the adjustment parameters a₁, a₂,a₃, and with states 0 to 3.

The minimum cumulative Euclidean distance from a convolutional coderwith one memory element, as illustrated in FIG. 3A or 3B for example, aswell as the distances just above, are caused by particular sequences inthe trellis which start from a given state and converge in anotherstate, in particular the shorter sequences. These sequences are called“DC” (for “Diverge and Converge”). Some examples of DC sequences areillustrated in FIG. 4 for a code defined on GF(4): an example of a DCsequence of length 2, an example of a DC sequence of length 3 and anexample of a DC sequence of length 4.

For a pair of DC sequences, X¹ and X², spanning L trellis sections (forexample sequences 411 and 412 spanning two trellis sections, thesequences 421 and 422 spanning three trellis sequences, or the sequences431 and 432 spanning four trellis sections), the cumulative Euclideandistance between the modulation symbols associated with these sequencesis calculated as follows:

$\begin{matrix}{d_{euc} = \sqrt{\sum\limits_{l = 1}^{L}\left\lbrack {{d^{2}\left( {X_{ls}^{m1},X_{ls}^{m2}} \right)} + {d^{2}\left( {X_{lp}^{m1},X_{lp}^{m2}} \right)}} \right\rbrack}} & \left\lbrack {{Equation}2} \right\rbrack\end{matrix}$ $\begin{matrix}{d_{euc} = \sqrt{\sum\limits_{l = 1}^{L}\left\lbrack {\left( {I_{X_{ls}^{m1}} - I_{X_{ls}^{m2}}} \right)^{2} + \left( {Q_{X_{ls}^{m1}} - Q_{X_{ls}^{m2}}} \right)^{2} + \text{ }\left( {I_{X_{lp}^{m1}} - I_{X_{lp}^{m2}}} \right)^{2} + \left( {Q_{X_{lp}^{m1}} - Q_{X_{lp}^{m2}}} \right)^{2}} \right\rbrack}} & \left\lbrack {{Equation}3} \right\rbrack\end{matrix}$where:X_(ls) ^(mb) and X_(lp) ^(mb) are respectively the modulation symbolsonto which are mapped the systematic and parity symbols corresponding tothe trellis section I in the sequence X^(b), b=1, 2 and I_(x) and Q_(x)represent the in-phase and quadrature components of the signal x in theconsidered constellation.

Due to the total connection property of the trellis, the pairs ofsystematic symbols S carried by the different sections of the DCsequences X¹ and X² can take all the combinations in GF(q)×GF(q). Morespecifically, the systematic symbols carried by the first and lasttrellis sections of the sequences DC are necessarily different on thetwo sequences. This is not the case for the symbols carried by theintermediate trellis sections when L>2, for which the systematic symbolsmay be identical.

The inventors of the present patent application have demonstrated thatfor this family of codes with v=1 memory element, the DC sequences thatare associated with modulation symbols whose cumulative Euclideandistance relative to the transmitted sequence is small, are essentiallygenerated by modulation symbols located in the vicinity of modulationsymbols actually transmitted, that is to say for which the followingterms in equation 3 are of low value:(I _(x) _(ls) _(m1) −I _(x) _(ls) _(m2) )²+(Q _(x) _(ls) _(m1) −Q _(x)_(ls) _(m2) )².

For example, if a recursive coder as illustrated in FIG. 3A, defined onGF(64), is considered, the inventors of the present patent applicationhave demonstrated that regardless of the values of the coefficientsa_(i), the two terms of the lowest cumulative Euclidean distances of theUB are obtained for pairs of sequences wherein the modulation symbolsassociated with systematic symbols are spaced two by two by at most thedistance d₁ indicated in FIG. 5 for a 64-QAM constellation. In otherwords, for a systematic symbol transmitted on a given modulation symbol,for example the modulation symbol 12 (whose binary representation is001100) in the constellation, the most probable concurrent systematicsymbols are those located in the circle of radius d₁ centred on themodulation symbol 12 in the constellation.

5.3 Examples of Implementation

In order to move the information symbols transmitted away from the mostprobable concurrent information symbols, and therefore to increase theminimum Euclidean distance between two code words at the output of theturbo-coder, it is proposed according to the invention to introduce atransformation before one of the encoding steps implemented by theturbo-encoder.

In particular, the transformation maximises the cumulative gap betweenthe modulation symbols associated with the information symbols beforetransformation and after transformation. As a result, maximising thecumulative gap between the modulation symbols associated with theinformation symbols before transformation and after transformationallows to increase the minimum Euclidean distance between two code wordsat the output of the turbo-coder.

The cumulative gap between two information symbols, or more specificallybetween two modulation symbols associated with two information symbols,can in particular be expressed in terms of squared Euclidean distance.Of course, other types of distance can be used to measure the gap, ordistance, between information symbols.

In other words, if the dispersion Δ (S_(i), S_(j)) between twoinformation symbols S_(i) and S_(j) of GF(q) is defined as thecumulative distance between two information symbols before and aftertransformation by the function T:Δ(S _(i) ,S _(j))=D(S _(i) ,S _(j))+D(T(S _(i)),T(S _(j)))  [Equation 4]the transformation is selected so as to maximise the minimum value ofthe dispersion Δ(S_(i), S_(j)), denoted Δ_(min), or at the very least toobtain a minimum value of the dispersion Δ (S_(i), S_(j)) greater than aselection threshold:

$\begin{matrix}{\Delta_{\min} = {\min\limits_{{({S_{i}S_{j}})} \in {{GF}(q)}^{2}}{\Delta\left( {S_{i},S_{j}} \right)}}} & \left\lbrack {{Equation}5} \right\rbrack\end{matrix}$

In particular, the dispersion Δ (S_(i), S_(j)) between two informationsymbols S_(i) and S_(j) of GF(q) is expressed as the sum of the squareof the Euclidean distance between the modulation symbols S_(i) ^(m),S_(j) ^(m) on which the information symbols S_(i), S_(j) are mappedbefore transformation and the square of the Euclidean distance betweenthe modulation symbols T(S_(i))^(m), T(S_(j))^(m) onto which theinformation symbols S_(i), S_(j) are mapped after transformation by thefunction T.Δ(S _(i) ,S _(j))=d _(euc) ²(S _(i) ^(m) ,S _(j) ^(m))+d _(euc) ²(T(S_(i))^(m) ,T(S _(j))^(m))  [Equation 6]

The expression of the dispersion given by equation 6 gives integervalues in the case of q-QAM constellations. However, any otherexpression of the dispersion whose maximisation or minimisation amountsto maximising the cumulative gap between the information symbols beforeand after transformation can be used.

The purpose of the transformation function is thus to ensure a highvalue of Δ_(min), that is to say greater than a selection threshold Th,in order to guarantee that if two information symbols are close (forexample mapped on neighbouring modulation symbols in the constellation)before transformation, they are distant after transformation, and viceversa.

As it can be tedious to enumerate all the transformations in order toselect the one which has the maximum value of Δ_(min), in particular forhigh values of q (cardinality of the Galois field), it is proposedaccording to the invention to select sufficiently large values ofΔ_(min), and for example greater than the selection threshold Th.

Various examples of transformation are presented below allowing toobtain sufficiently high minimum dispersion values Δ_(min), depending onwhether the cardinal of the Galois field on which the code is defined,q, and the order of the modulation, p, are identical or different.

A) Code and Modulation of the Same Order: q=p

When the order p of the modulation, represented by a constellation, isequal to the cardinal q of the Galois field to which the informationsymbols belong, each information or redundancy symbol is mapped to aunique modulation symbol. In other words, if a transmission chaincomprising a turbo-encoding step according to the invention isconsidered, taking as input information symbols defined on a Galoisfield GF(q) and delivering the information symbols and redundancysymbols defined on the Galois field GF(q), and a q-QAM type amplitudemodulation step, each information or redundancy symbol resulting fromthe turbo-coding step is transmitted on a modulation symbol of the q-QAMmodulation, each modulation symbol being defined on the alphabet q(q-ary symbol).

In this case, the dispersion Δ (S_(i), S_(j)) between two informationsymbols S_(i) and S_(j) of GF(q) is expressed as:Δ(S _(i) ,S _(j))=d _(euc) ²(S _(i) ^(m) ,S _(j) ^(m))+d _(euc) ²(T(S_(i))^(m) ,T(S _(j))^(m))with:

-   -   d_(euc) ²(S_(i) ^(m),S_(j) ^(m))=(I_(s) _(i) _(m) I_(s) _(j)        _(m) )²+(Q_(s) _(i) _(m) −Q_(s) _(j) _(m) )² the square of the        Euclidean distance between the modulation symbols S_(i) ^(m),        S_(j) ^(m) onto which said information symbols S_(i), S_(j) are        mapped before transformation;    -   d_(euc) ²(T(S_(i))^(m),T(S_(j))^(m))=(I_(T(S) _(i) ₎ _(m)        −I_(T(S) _(j) ₎ _(m) )²+(Q_(T(S) _(i) ₎ _(m) −Q_(T(S) _(j) ₎        _(m) )² the square of the Euclidean distance between the        modulation symbols T(S_(i))^(m), T(S_(j))^(m) onto which said        information symbols S_(i), S_(j) are mapped after        transformation; and    -   I_(x) and Q_(x) are the in-phase and quadrature components of a        signal x in the considered constellation.

According to a first example of implementation, the transformationtransforms a pair of information symbols intended to be mapped onto apair of modulation symbols whose Euclidean distance is less than a firstthreshold, into a pair of information symbols intended to be mapped ontoa pair of modulation symbols whose Euclidean distance is greater thanthe first threshold.

In other words, if the pair of information symbols (S_(i), S_(j)) beforetransformation is mapped onto a pair of modulation symbols having asmall Euclidean distance, the pair of information symbols (T (S_(i)), T(S_(j))) after transformation must be mapped onto a modulation symbolpair having a large Euclidean distance, and vice versa, so as tomaximise the minimum dispersion between the information symbols S_(i),S_(j).

An example of the construction of the transformation function ispresented below which allows, according to a first criterion, toguarantee a minimum value for the dispersion between the informationsymbols S_(i), S_(j).

According to this example, in order to guarantee a minimum dispersionfor each pair of information symbols (S_(i), S_(j)), it is necessary toknow the distribution of the Euclidean distances between the set ofpairs of modulation symbols associated with the set of pairs ofinformation symbols in the considered constellation. For a 64-QAMconstellation as illustrated in FIG. 5 for example, 33 different valuesof distances between two different modulation symbols can be listed.

The table below provides the list of multiplicities m(D_(i)) associatedwith each distance D_(i), that is to say the number of pairs ofmodulation symbols at each distance D_(i), the distances beingclassified in increasing order. D₁ corresponds to the distance betweentwo direct neighbours and D₃₃ to the distance between two oppositecorners of the constellation:

TABLE 2 D_(i) m(D_(i)) D_(i) m(D_(i)) D_(i) m(D_(i)) D₁ 112 D₁₂ 96 D₂₃16 D₂ 98 D₁₃ 128 D₂₄ 46 D₃ 96 D₁₄ 84 D₂₅ 32 D₄ 168 D₁₅ 72 D₂₆ 24 D₅ 72D₁₆ 32 D₂₇ 20 D₆ 80 D₁₇ 60 D₂₈ 24 D₇ 140 D₁₈ 32 D₂₉ 16 D₈ 120 D₁₉ 56 D₃₀8 D₉ 64 D₂₀ 48 D₃₁ 12 D₁₀ 112 D₂₁ 48 D₃₂ 8 D₁₁ 50 D₂₂ 40 D₃₃ 2

A first rule for constructing the transformation function consists inassociating the pairs whose distances before transformation are at thestart of the table with pairs whose distances after transformation areat the end of the table.

For example, a maximum value of i, denoted i_(max), is sought such thatif the Euclidean distance before transformation between all the pairs ofmodulation symbols associated with the information symbols S and S′,denoted d_(euc) (S^(m), S′^(m)), is less than or equal to Di_(max),their distance after transformation, denoted d_(euc) (T(S)^(m),T(S′)^(m)), is strictly greater than Di_(max). Then, this guaranteesΔ_(min)≥D_(i) _(max) ₊₁+D₁.

For example, for a 64-QAM modulation as shown in FIG. 5 , i_(max) isequal to 3: Di_(max)=D₃.

Table 3 below provides the list in multiplicities associated withEuclidean distances after transformation for pairs of informationsymbols S and S′ whose Euclidean distance before transformation is lessthan Di_(max):d_(euc)(S^(m),S^(nm))≤D_(i) _(max) .

It can be seen that all the pairs of modulation symbols associated withinformation symbols have a Euclidean distance strictly greater thanDi_(max)=D₃, since the first three multiplicity values are equal to 0.

TABLE 3 D_(i) m(D_(i)) D_(i) m(D_(i)) D_(i) m(D_(i)) D₁ 0 D₁₂ 15 D₂₃ 5D₂ 0 D₁₃ 29 D₂₄ 12 D₃ 0 D₁₄ 6 D₂₅ 7 D₄ 25 D₁₅ 7 D₂₆ 3 D₅ 16 D₁₆ 5 D₂₇ 8D₆ 21 D₁₇ 12 D₂₈ 4 D₇ 27 D₁₈ 3 D₂₉ 6 D₈ 19 D₁₉ 8 D₃₀ 2 D₉ 10 D₂₀ 5 D₃₁ 1D₁₀ 22 D₂₁ 3 D₃₂ 2 D₁₁ 13 D₂₂ 9 D₃₃ 1

According to a second example of implementation, the transformationtransforms an information symbol intended to be mapped onto a modulationsymbol having, in the constellation, a number of neighbouring modulationsymbols less than a determined number, into an information symbolintended to be mapped onto a modulation symbol having, in theconstellation, a number of neighbouring modulation symbols greater thansaid determined number, and vice versa, as long as this is possible.

An example of the construction of the transformation function ispresented below, allowing, according to a second criterion, to optimisethe level of protection of information symbols.

Considering, as an example, the case of the 64-QAM constellationillustrated in FIG. 6 , it is seen that the 28 modulation symbolsrepresented by the constellation points located at the corners (circledpoints) and on the edges (boxed points) of the constellation have fewerdirect neighbours than the modulation symbols located at the centre ofthe constellation. The constellation therefore gives the information orredundancy symbols mapped to these modulation symbols located at thecorners and on the edges better protection, because the number ofmodulation symbols with which these modulation symbols located at thecorners and on the edges can coincide is lower.

Consequently, if the transformation is constructed such that theinformation symbols mapped onto modulation symbols located on the edgesand corners of the constellation before (respectively after)transformation end up in the central part of the constellation after(respectively before) transformation, the total number of informationsymbols benefiting from this increased protection will be doubledcompared to a coding pattern without transformation.

In other words, in the case where the information and redundancy symbolsfrom the turbo-coder are associated with a constellation havingdifferent levels of symbol protection, a second rule of construction ofthe transformation can thus be defined.

Note that these two examples of implementation of the transformation canbe applied independently (that is to say individually) or incombination.

For example, two transformation functions, denoted T1 and T2, arepresented, said functions are obtained by applying the two precedingconstruction rules, and according to which the information andredundancy symbols belong to the Galois field GF(64) and are intended tobe mapped onto modulation symbols of a 64-QAM quadrature amplitudemodulation as shown in FIG. 5 .

The mapping of symbols of GF(64) onto the 64-QAM constellation is givenby the following table, where b₅b₄b₃b₂b₁b₀ is the binary representationof each symbol in GF(64), b₅ representing the most significant bit (MSB)and b₀ the least significant bit (LSB).

TABLE 4 Value of Q b₅b₃b₁ Value of I b₄b₂b₀ +7 000 +7 000 +5 001 +5 001+3 011 +3 011 +1 010 +1 010 −1 110 −1 110 −3 111 −3 111 −5 101 −5 101 −7100 −7 100

The transformation T1 corresponds to a minimum dispersion valueΔ_(min)=12, and the transformation T2 corresponds to the largest minimumdispersion value found for a 64-QAM constellation, Δ_(min)=28:

TABLE 5 x T1(x) T2(x) x T1(x) T2(x) x T1(x) T2(x) x T1(x) T2(x) 0 0 1516 48 7 32 13 11 48 61 41 1 3 19 17 51 25 33 14 47 49 62 62 2 6 29 18 5439 34 11 28 50 59 31 3 5 0 19 53 9 35 8 50 51 56 33 4 12 3 20 60 51 36 163 52 49 45 5 15 35 21 63 46 37 2 27 53 50 22 6 10 12 22 58 37 38 7 1054 55 49 7 9 4 23 57 17 39 4 5 55 52 18 8 24 6 24 40 14 40 21 57 56 3743 9 27 8 25 43 2 41 22 44 57 38 60 10 30 38 26 46 59 42 19 54 58 35 5511 29 32 27 45 52 43 16 21 59 32 56 12 20 1 28 36 34 44 25 58 60 41 1313 23 16 29 39 26 45 26 53 61 42 20 14 18 23 30 34 61 46 31 30 62 47 3615 17 24 31 33 48 47 28 42 63 44 40

According to a third example of implementation, the method forgenerating a signal implements a step of selecting the transformationfrom several available transformations, the selection taking intoaccount the redundancy symbols and/or the interleaver.

In particular, in the case where the first and/or second rules allow toconstruct several equivalent transformation functions according to thecorresponding criteria, a third criterion can be applied, to select oneof the transformations.

For this purpose, for each sequence X comprising two or three systematicsymbols on GF(q), the concurrent sequences X′ comprising the mostprobable concurrent systematic symbols are identified, as definedpreviously and illustrated in FIG. 5 for GF(64). Indeed, as demonstratedpreviously, it is the DC sequences of length 2 or 3 which are at theorigin of the minimum Euclidean distance of the coded modulation, whichlimits the performances at low error rates.

The cumulative Euclidean distance of the modulation symbols associatedwith the redundancy symbols between each transmitted sequence and theconcurrent sequences is then determined, before and aftertransformation:

$\begin{matrix}{d_{{euc},p}^{2} = {\sum\limits_{l = 1}^{L}\left\lbrack {{d^{2}\ \left( {X_{lp}^{m},X_{lp}^{m^{\prime}}} \right)} + {d^{2}\left( {Y_{lp}^{m},\ Y_{lp}^{m^{\prime}}} \right)}} \right\rbrack}} & \left\lbrack {{Equation}7} \right\rbrack\end{matrix}$where:

L is the number of modulation symbols associated with systematic symbols(that is to say L=2 or L=3);

X_(lp) ^(m),X_(lp) ^(m)′ represent respectively the modulation symbolsassociated with the I-th parity symbols of the sequences X and X′obtained from the information symbols;

Y_(lp) ^(m),Y_(lp) ^(m)′ represent respectively the modulation symbolsassociated with the I-th parity symbols of the sequences T(X) and T(X′)obtained from the transformed information symbols;

The selection step selects for example the transformation T which leadsto the best distance spectrum, where the distance spectrum is composedof all the calculated distance values sorted in ascending orderassociated with the number of occurrences (multiplicity) for eachdistance. The best distance spectrum is the one that gives the greatestminimum distance. If the minimum distances are equal, for example, thespectrum with a lower number of occurrences of the minimum distance isselected. If the multiplicity is equal, the distance value just abovethe minimum distance can be checked.

According to this third example of implementation, the most advantageoustransformation in terms of cumulative Euclidean distance is thusselected.

It is noted that if the transformation function has two components, oneimplemented before the first encoding and the other before the secondencoding, before or after interleaving, the above criteria apply so asto optimise the overall transformation function.

B) Code and Modulation of Different Order: q>p

When the modulation order p, represented by a constellation, is lessthan the cardinal q of the Galois field to which the information symbolsbelong, each information or redundancy symbol is mapped onto severalmodulation symbols. In other words, if a transmission chain comprising aturbo-encoding step according to the invention is considered, taking asinput information symbols defined on a Galois field GF(q), anddelivering the information symbols and redundancy symbols defined on theGalois field GF(q), and a modulation step of order p, each informationor redundancy symbol resulting from the turbo-coding step is transmittedon n modulation symbols, each modulation symbol being defined on thealphabet p (p-ary symbol).

The introduction of a transformation function thus allows to improve theperformances of correction at a low error rate when the symbols of theGalois field on which the code, q, is defined, are transmitted using nsymbols of a constellation with a lower order p: q=p^(n).

For example, each information or redundancy symbol derived from theturbo-coder, defined on GF(256), is transmitted on two modulationsymbols of a 16-QAM modulation (q=256, p=16, n=2), defined in thealphabet 0 to 15, where each information or redundancy symbol from theturbo-coder, defined on GF(64), is transmitted on three QPSK modulationsymbols (q=64, p=4, n=3), defined in the alphabet 0 to 3.

In this case, each information or redundancy symbol of GF(q) can berepresented by a set of n modulation symbols associated with aconstellation of order p (for example p-QAM).

In this case, the dispersion Δ (S_(i), S_(j)) between two informationsymbols S_(i) and S_(j) of GF(q) is expressed as:

$\begin{matrix}{{{\Delta\left( {S_{i},S_{j}} \right)} = {{{D\left( {S_{i},S_{j}} \right)} + {D\left( {{T\left( S_{i} \right)},\ {T\left( S_{j} \right)}} \right)}} = {{\Delta\left( {S_{i},S_{j}} \right)} = {{{\overset{n}{\sum\limits_{k = 1}}{d_{euc}^{2}\left( {S_{i,k}^{m},S_{j,k}^{m}} \right)}} + {\overset{n}{\sum\limits_{k = 1}}{d_{euc}^{2}\left( {{T\left( S_{i} \right)}_{k}^{m},{T\left( S_{j} \right)}_{k}^{m}} \right)}}} = {{\overset{n}{\sum\limits_{k = 1}}\left\lbrack {\left( {I_{S_{i,k}^{m}} - I_{S_{j,k}^{m}}} \right)^{2} + \left( {Q_{S_{i,k}^{m}} - Q_{S_{j,k}^{m}}} \right)^{2}} \right\rbrack} + {\sum\limits_{k = 1}^{n}\left\lbrack {\left( {I_{{T(S_{i})}_{k}^{m}} - I_{{T(S_{j})}_{k}^{m}}} \right)^{2} + \left( {Q_{{T(S_{i})}_{k}^{m}} - Q_{{T(S_{j})}_{k}^{m}}} \right)^{2}} \right\rbrack}}}}}}} & \left\lbrack {{Equation}8} \right\rbrack\end{matrix}$with:

-   -   S^(m) _(i,k) respectively S^(m) _(j,k), for k ranging from 1 to        n, the n modulation symbols associated with the information        symbol S_(i), respectively S_(j), before transformation,        T(S_(i))_(k) ^(m), respectively T(S_(j))_(k) ^(m), for k ranging        from 1 to n, the n modulation symbols associated with the        information symbol S_(i), respectively S_(j), after        transformation, I_(x) and Q_(x) are the in-phase and quadrature        components of a signal x in the considered constellation.

The distance term between two symbols of GF(q) therefore breaks downinto n terms, each of the n terms representing a Euclidean distance inthe space of the constellation of order p.

Various implementation examples are presented below to obtain asufficiently high minimum dispersion value Δ_(min) that is to saygreater than a selection threshold, in this particular case where q>p:

$\Delta_{\min} = {\min\limits_{{({S_{i}S_{j}})} \in {G{F(q)}^{2}}}{{\Delta\left( {S_{i},S_{j}} \right)}.}}$

According to a first example of implementation, the transformationminimises the number of zero terms in equation 8 of the dispersionΔ(S_(i),S_(j)).

More specifically, each expression of the distance D(S_(i),S_(j))contains n terms of Euclidean distance in the space of a constellationwith p signals. consequently, it is sought to minimise the number ofp-ary symbols, among the n modulation symbols associated with eachinformation symbol of GF(q), which are identical before and aftertransformation.

For this purpose, the transformation associates pairs of informationsymbols of GF(q) having many p-ary symbols in common, with pairs ofinformation symbols having few p-ary symbols in common, and vice versa.

In other words, the transformation transforms a pair of informationsymbols each intended to be mapped onto n p-ary modulation symbols, thenumber of p-ary symbols in common between the two sets of n p-arymodulation symbols being greater than a second threshold, into a pair ofinformation symbols each intended to be mapped onto n p-ary modulationsymbols, the number of p-ary symbols in common between these two sets ofn p-ary modulation symbols being less than the second threshold, andvice versa.

For example, if a turbo-code defined on the Galois field GF(64) isconsidered, for which each information or redundancy symbol istransmitted on three QPSK modulation symbols (that is to say q=64, p=4,n=3), two distinct information symbols in GF(64) may differ by one, twoor three distinct modulation symbols in the QPSK constellation.

Thus, if the information symbol 19 (010011) defined in GF(64) isintended to be mapped onto the three QPSK modulation symbols 103 and theinformation symbol 62 (111110) is intended to be mapped onto the threemodulation QPSK symbols 332, the two information symbols 19 and 62differ, in the QPSK constellation, by three modulation symbols (1≠3,0≠3, 3≠2). The expression of the distance between the informationsymbols, D (19,62) contains no zero term.

If the information symbol 55 (110111) is intended to be mapped onto thethree modulation symbols 313, and the information symbol 62 (111110) isintended to be mapped onto the three modulation symbols 332, the twoinformation symbols differ, in the QPSK constellation, by two modulationsymbols (3=3, 1≠3, 3≠2). The expression of the distance between theinformation symbols, D (55,62) therefore contains one zero term.

If the information symbol 63 (111111) is intended to be mapped onto thethree modulation symbols 333, and the information symbol 62 (111110) isintended to be mapped onto the three modulation symbols 332, the twoinformation symbols differ, in the QPSK constellation, by a singlemodulation symbol (3=3, 3=3, 3≠2). The expression of the distancebetween the information symbols, D (63,62) therefore contains two zeroterms.

Among the C₆₄ ²=2016 possible combinations of pairs of informationsymbols of GF(64), we determine as follows:

-   -   864 pairs having no QPSK symbol in common when the information        symbols are represented in the QPSK constellation,    -   864 pairs having one QPSK symbol in common when the information        symbols are represented in the QPSK constellation,    -   288 pairs having two QPSK symbols in common when the information        symbols are represented in the QPSK constellation.

The proposed transformation allows to associate as a priority the pairsof information symbols having, in the QPSK constellation, two QPSKsymbols in common with the pairs of information symbols having, in theQPSK constellation, no QPSK symbol in common, then the pairs ofinformation symbols having, in the QPSK constellation, one QPSK symbolin common with the pairs of information symbols having, in the QPSKconstellation, no QPSK symbol in common.

For example, the transformation T3 proposed below in Table 7 allows totransform the 288 pairs of information symbols having two QPSK symbolsin common, before transformation, into 288 pairs having no QPSK symbolin common, after transformation, and to transform 48 pairs ofinformation symbols having one QPSK symbol in common, beforetransformation, into 48 pairs having no QPSK symbol in common, aftertransformation. The remaining 864-48=816 pairs of information symbolskeep one common QPSK symbol after transformation.

The mapping to the QPSK constellation is given by the following Table 6,where b₁b₀ is the binary representation associated with each QPSKmodulation symbol, b₁ representing the most significant bit (MSB) and b₀the least significant bit (LSB).

TABLE 6 Value of Q b₁ Value of I b₀ +1 0 +1 0 −1 1 −1 1

TABLE 7 x T3(x) x T3(x) x T3(x) x T3(x) 0 0 16 30 32 39 48 57 1 21 17 1133 50 49 44 2 42 18 52 34 13 50 19 3 63 19 33 35 24 51 6 4 27 20 5 36 6052 34 5 14 21 16 37 41 53 55 6 49 22 47 38 22 54 8 7 36 23 58 39 3 55 298 45 24 51 40 10 56 20 9 56 25 38 41 31 57 1 10 7 26 25 42 48 58 46 1118 27 12 43 37 59 59 12 54 28 40 44 17 60 15 13 35 29 61 45 4 61 26 1428 30 2 46 43 62 53 15 9 31 23 47 62 63 32

According to a second example of implementation, the method forgenerating a signal according to the invention comprises a step ofselecting the transformation from several available transformations, theselection maximising the non-zero terms in equation 8 of the dispersionΔ(S_(i),S_(j)).

In particular, in the case where the example of implementation proposedpreviously allows to construct several equivalent transformationfunctions, a selection step can be implemented, allowing to select thetransformation function which maximises the non-zero values of the terms

[(I_(S_(i, k)^(m)) − I_(S_(j, k)^(m)))² + (Q_(S_(i, k)^(m)) − Q_(S_(j, k)^(m)))²] + [⁠(I_(T(S_(i))_(k)^(m)) − I_(T(S_(j))_(k)^(m)))² + (Q_(T(S_(i))_(k)^(m)) − Q_(T(S_(j))_(k)^(m)))²]

Such a criterion indeed allows to guarantee that p-ary symbols which areclose before transformation will be distant after transformation.

5.4 Performances

Some simulation results are presented below, allowing to measure theeffect of the introduction of a transformation on the correctionperformances of a turbo-code, independently of the effect of theinterleaving.

FIGS. 7A to 7C show the error rate (FER) as a function of thesignal-to-noise ratio (E_(b)/N₀) for different simulations, when theorder of modulation and turbo-code are the same.

To obtain these results, we consider:

-   -   the structure of FIG. 2A, according to which the information        symbols entering the first encoder C1 21A are neither        interleaved nor transformed, and the information symbols        entering the second encoder C2 22A are interleaved, then        transformed;    -   each coder C1 21A, C2 22A implements the same elementary code        with a single memory element, as illustrated in FIG. 3A;    -   the efficiency of the turbo-code is ⅓;    -   the length of digital input messages is K=900 information        symbols;    -   the information symbols are defined on GF(64);    -   the modulation is 64-QAM modulation;    -   the length of the digital input messages expressed in bits is        K_(b)=K×6=5400 bits of information.

For the results illustrated in FIGS. 7A and 7B, the elementary codeadjustment parameters are for example: a₁=31, a₂=5 and a₃=18, and forthe results illustrated in FIG. 7C, the elementary code adjustmentparameters are for example: a₁=41, a₂=2 and a₃=0, and the code isconstructed using the primitive polynomial P(D)=1+D²+D³+D⁵+D⁶.

Different transformations have been tested:

-   -   configuration 1: the transformation is the identity function        (there is no transformation). In this case, for a 64-QAM        modulation, we have:

$\Delta_{\min} = {{\underset{{({S_{i},S_{j}})} \in {G{F(q)}^{2}}}{2 \times \min}{d_{euc}^{2}\left( {S_{i}^{m},S_{j}^{m}} \right)}} = 8}$

-   -   configuration 2: the transformation T1 corresponds to a minimum        dispersion value Δ_(min)=12;    -   configuration 3: the transformation T2 corresponds to a minimum        dispersion value Δ_(min)=28, that is to say the greatest value        of Δ_(min) found for a 64-QAM constellation.

The transformations T1 and T2 for the 64-QAM constellation in FIG. 5 areshown in Table 5.

A first simulation consisted of simulating a turbo-code whoseinterleaving function is the identity function. In other words, theinformation symbol sequences before and after interleaving areidentical. In this case, no performance gain associated withinterleaving (“interleaving gain”) is expected.

The results of this first simulation are illustrated in FIG. 7A.

When no transformation is applied (curve 71A), the correctionperformances of the turbo-code are similar to those of the elementaryconvolutional code for which the transmission of the parity symbols isrepeated.

When a transformation is inserted before coding by the second encoder C222A, for example the transformation T1 (curve 72A) or the transformationT2 (curve 72B), a coding gain is observed, which increases with thevalue of the minimum dispersion Δ_(min).

A second simulation consisted of simulating a turbo-code whoseinterleaving function is a random function (also called uniforminterleaving). This is a probabilistic interleaver that allows toestimate the average turbo-code interleaving gain, independently of aparticular interleaving pattern. The interleaver is randomly drawn foreach message transmitted. The error rate curves thus obtained representthe performances of the code averaged over all the possibleinterleavers.

The results of this second simulation are illustrated in FIG. 7B.

Unlike the performances of FIG. 7A, here the presence of a coding gainlinked to the interleaving can be observed. Also the error floorphenomenon is observed, which is characteristic of turbo-codes when theinterleaving is not optimised. The position of the error floor on curves72A (no transformation), 72B (with transformation T1) or 72C (withtransformation T2) is given by the union bound of the turbo-code,according to the equation 1.

It can be seen in FIG. 7B that the introduction of a transformationallows to lower the error floor and to improve the correction power at alow error rate of the turbo-code, and especially when the minimumdispersion parameter Δ_(min) associated with the transformation is high.This confirms the increase in the minimum cumulative Euclidean distanceof the turbo-code with the maximisation of the minimum dispersionparameter Δ_(min).

A third simulation consisted in simulating a turbo-code whoseinterleaving function is a conventional interleaving function, forexample of the ARP type (“Almost Regular Interleaver”).

The results of this third simulation are illustrated in FIG. 7C.

The same phenomena as in FIG. 7B are observed in FIG. 7C.

In particular, as the parameters a1=41, a2=2 and a3=0 of each elementarycode (C1, C2) have been modified, the turbo-code obtained does not havegood distance properties and has a floor error rate which is naturallyhigh without transformation (curve 71C). It can be observed that the useof the transformation T2, in terms of dispersion, allows this floorerror rate to be lowered by about 2 decades (curves 73C).

A similar phenomenon has also been observed for other elementary codeson GF(64) with the same transformations, as well as for codes defined onGF(16), associated with a 16-QAM modulation.

FIG. 8 illustrates the error rate (FER) as a function of thesignal-to-noise ratio (E_(b)/N₀) for different simulations, when theorder of modulation and turbo-code are different.

To obtain these results, we consider:

-   -   the structure of FIG. 2A, according to which the information        symbols entering the first encoder C1 21A are neither        interleaved nor transformed, and the information symbols        entering the second encoder C2 22A are interleaved, then        transformed;    -   each encoder C1 21A, C2 22A implements the same elementary code        with a single memory element, as illustrated in FIG. 3A;    -   the length of the digital input messages is K=900 information        symbols;    -   the information symbols are defined on GF(64);    -   the modulation is a QPSK modulation;    -   the length of the digital input messages expressed in bits is        K_(b)=K×6=5400 information bits.

For the results illustrated in FIG. 8 , the adjustment parameters of theelementary code are a₁=31, a₂=5 and a₃=18 and the code is constructedusing the primitive polynomial P(D)=1+D²+D³+D⁵+D⁶.

Different transformations have been tested:

-   -   configuration 1: the transformation is the identity function        (there is no transformation);    -   configuration 2: transformation T3 corresponds to a minimum        dispersion value Δ_(min)=16.

The transformation T3 is described in Table 7.

The turbo-code thus obtained was simulated for a transmission on aGaussian channel, using a random interleaver. FIG. 8 shows theperformance curves without implementation of a transformation (curve 81)and with implementation of the transformation T3 (curve 82), and it canbe seen that the introduction of the transformation also allows in thiscontext to significantly lower the error floor and improve thecorrection power at a low error rate of the turbo-code.

It is noted that in the above simulations, it was considered that theencoders C1 and C2 implement the same elementary code, for example thatillustrated in FIG. 3A. Of course, these encoders can implementdifferent codes, and in particular codes having different correctivepower. In particular, only one systematic output is provided in theturbo-encoder.

5.5 Structure

Finally, in relation to FIG. 9 , the simplified structure of aturbo-coder according to at least one embodiment described above ispresented.

Such a turbo-encoder comprises at least one memory 91 comprising abuffer memory, at least one processing unit 92, equipped for examplewith a programmable computing machine or with a dedicated computingmachine, for example a processor P, and controlled by the computerprogram 93, implementing steps of the method for generating a signalaccording to at least one embodiment of the invention.

Upon initialisation, the code instructions of the computer program 93are for example loaded into a RAM memory before being executed by theprocessor of the processing unit 92.

The processor of the processing unit 92 implements steps of the methodof generating a signal described above, according to the instructions ofthe computer program 93, to:

-   -   code a set of information symbols (either the input information        symbols or the transformed information symbols) using a first        code, delivering a first set of redundancy symbols,    -   interleave the set of information symbols (either the input        information symbols or the transformed information symbols),        delivering a set of interleaved information symbols,    -   code the set of interleaved information symbols (either the        input information symbols or the transformed information        symbols) using a second code, delivering a second set of        redundancy symbols,    -   apply a bijective transformation to the information symbols        (either the input information symbols or the interleaved        information symbols), before and/or after the interleaving.

The invention claimed is:
 1. A coding device comprising: a turbo-encoderconfigured to receive an input digital message comprising a set ofinformation symbols and configured to implement a non-binary turbo-codetype error correcting code to encode the set of information symbolscomprised in the input digital message, the information symbolsbelonging to a Galois field of cardinal q, denoted GF(q), with q>2, saidturbo-encoder comprising: a first encoder configured to code said set ofinformation symbols and generate a first set of redundancy symbols; aninterleaver configured to interleave said set of information symbols andgenerate a set of interleaved information symbols; a second encoderconfigured to code said set of interleaved information symbols andgenerate a second set of redundancy symbols; and a processor configuredto: map the information and redundancy symbols onto modulation symbolsassociated with a constellation of order p, an information or redundancysymbol being mapped onto n modulation symbols, with n n≥2, the order pof the constellation being less than the cardinal q of the Galois fieldto which said information symbols belong; apply a bijectivetransformation to said information symbols before and/or after saidinterleaver, said transformation modifying the value of at least two ofsaid information symbols prior to the encoding of said informationsymbols by the first encoder and/or the second encoder, the bijectivetransformation being defined such that every input of a set of qelements of GF(q) has a unique output in the set of q elements of GF(q)by the bijective transformation and every output of the set of qelements of GF(q) has a unique antecedent in the set of q elements ofGF(q), said transformation taking into account said constellation;generate an output signal representing a code word which comprises theset of information symbols and the first and second sets of redundancysymbols; and transmit the output signal in a communication system.
 2. Anon-transitory computer-readable medium comprising instructions storedtherein, which when executed by a processor of a coding device,configure the coding device to: receive an input digital messagecomprising a set of information symbols; turbo-code the set ofinformation symbols comprised in the input digital message, theinformation symbols belonging to a Galois field of cardinal q, denotedGF(q), with q>2, wherein the turbo-coding implements a non-binaryturbo-code type error correcting code to obtain redundancy symbols andcomprises: encoding said set of information symbols by a first encoderof the coding device to generate a first set of redundancy symbols;interleaving said set of information symbols using an interleaver of thecoding device to generate a set of interleaved information symbols;encoding said set of interleaved information symbols by a second encoderof the coding device to generate a second set of redundancy symbols;mapping the information and redundancy symbols onto modulation symbolsassociated with a constellation of order p, an information or redundancysymbol being mapped onto n modulation symbols, with n≥2, the order p ofthe constellation being less than the cardinal q of the Galois field towhich said information symbols belong; applying a bijectivetransformation to said information symbols using a processor of thecoding device before and/or after said interleaving, said transformationmodifying a value of at least two of said information symbols prior tothe coding of said information symbols by the first encoder and/or thesecond encoder, the bijective transformation being defined such thatevery input of a set of q elements of GF(q) has a unique output in theset of q elements of GF(q) by the bijective transformation and everyoutput of the set of q elements of GF(q) has a unique antecedent in theset of q elements of GF(q), said transformation taking into account saidconstellation; generating an output signal representing a code wordwhich comprises the set of information symbols and the first and secondsets of redundancy symbols; and transmitting the output signal in acommunications system.
 3. A method performed by a coding device, themethod comprising: receiving an input digital message comprising a setof information symbols; turbo-coding the set of information symbolscomprised in the input digital message, the information symbolsbelonging to a Galois field of cardinal q, denoted GF(q), with q>2,wherein the turbo-coding implements a non-binary turbo-code type errorcorrecting code to obtain redundancy symbols and comprises: encodingsaid set of information symbols by a first encoder of the coding deviceto generate a first set of redundancy symbols, interleaving said set ofinformation symbols using an interleaver of the coding device togenerate a set of interleaved information symbols; encoding said set ofinterleaved information symbols by a second encoder of the coding deviceto generate a second set of redundancy symbols; and applying a bijectivetransformation to said information symbols using a processor of thecoding device before and/or after said interleaving, said transformationmodifying a value of at least two of said information symbols prior tothe coding of said information symbols by the first encoder and/or thesecond encoder, the bijective transformation being defined such thatevery input of a set of q elements of GF(q) has a unique output in theset of q elements of GF(q) by the bijective transformation and everyoutput of the set of q elements of GF(q) has a unique antecedent in theset of q elements of GF(q); generating an output signal representing acode word which comprises the set of information symbols and the firstand second sets of redundancy symbols; and transmitting the outputsignal in a communications system, wherein: said transformationgenerates a minimum dispersion Δ_(min) between two information symbolsS_(i), S_(j) among said information symbols greater than a selectionthreshold, such that:Δ_(min)=min Δ(S _(i) ,S _(j)) with:Δ(S_(i),S_(j))=D(S_(i),S_(j))+D(T(S_(i)),T(S_(j))) D(S_(i),S_(j)) beinga distance between said information symbols S_(i), S_(j) beforetransformation, D(T(S_(i)),T(S_(j))) being a distance between saidinformation symbols S_(i), S_(j) after transformation by the function T;the method comprises mapping the information and redundancy symbols ontomodulation symbols associated with a constellation of order p equal tothe cardinal q of the Galois field to which said information symbolsbelong, an information or redundancy symbol being mapped onto a singlemodulation symbol, and said transformation takes into account saidconstellation; and dispersion between two information symbols S_(i),S_(j) is expressed in the form:Δ(S _(i) ,S _(j))=d _(euc) ²(S _(i) ^(m) ,S _(j) ^(m))+d _(euc) ²(T(S_(i))^(m) ,T(S _(j))^(m)) with: d_(euc) ²(S_(i) ^(m),S_(j) ^(m))=(I_(s)_(i) _(m) −I_(s) _(j) _(m) )²+(Q_(s) _(i) _(m) −Q_(s) _(j) _(m) )² beingthe square of the Euclidean distance between the modulation symbolsS_(i) ^(m), S_(j) ^(m) onto which said information symbols S_(i), S_(j)are mapped before transformation; d_(euc)²(T(S_(i))^(m),T(S_(j))^(m))=(I_(T(S) _(i) ₎ _(m) −I_(T(S) _(j) ₎ _(m))²+(Q_(T(S) _(i) ₎ _(m) −Q_(T(S) _(j) ₎ _(m) )² being the square of theEuclidean distance between the modulation symbols T(S_(i))^(m),T(S_(j))^(m) onto which said information symbols S_(i), S_(j) are mappedafter transformation; I_(x) and Q_(x) are the in-phase and quadraturecomponents of a signal x in the considered constellation.
 4. The methodaccording to claim 3, wherein the said transformation transforms a pairof information symbols intended to be mapped onto a pair of modulationsymbols whose Euclidean distance is less than a first threshold, into apair of information symbols intended to be mapped onto a pair ofmodulation symbols whose Euclidean distance is greater than said firstthreshold.
 5. The method according to claim 3, wherein saidtransformation transforms an information symbol intended to be mappedonto a modulation symbol having, in the constellation, a number ofneighbouring modulation symbols less than a determined number, into aninformation symbol intended to be mapped onto a modulation symbolhaving, in the constellation, a number of neighbours greater than saiddetermined number.
 6. A method performed by a coding device, the methodcomprising: receiving an input digital message comprising a set ofinformation symbols; turbo-coding the set of information symbolscomprised in the input digital message, the information symbolsbelonging to a Galois field of cardinal q, denoted GF(q), with q>2,wherein the turbo-coding implements a non-binary turbo-code type errorcorrecting code to obtain redundancy symbols and comprises: encodingsaid set of information symbols by a first encoder of the coding deviceto generate a first set of redundancy symbols, interleaving said set ofinformation symbols using an interleaver of the coding device togenerate a set of interleaved information symbols; encoding said set ofinterleaved information symbols by a second encoder of the coding deviceto generate a second set of redundancy symbols; and applying a bijectivetransformation to said information symbols using a processor of thecoding device before and/or after said interleaving, said transformationmodifying a value of at least two of said information symbols prior tothe coding of said information symbols by the first encoder and/or thesecond encoder, the bijective transformation being defined such thatevery input of a set of q elements of GF(q) has a unique output in theset of q elements of GF(q) by the bijective transformation and everyoutput of the set of q elements of GF(q) has a unique antecedent in theset of q elements of GF(q); generating an output signal representing acode word which comprises the set of information symbols and the firstand second sets of redundancy symbols; and transmitting the outputsignal in a communications system, wherein: the method comprises mappingthe information and redundancy symbols onto modulation symbolsassociated with a constellation of order p, an information or redundancysymbol being mapped onto n modulation symbols, with n≥2, the order p ofthe constellation being less than the cardinal q of the Galois field towhich said information symbols belong; and said transformation takesinto account said constellation.
 7. The method according to the claim 6,wherein said transformation generates a minimum dispersion Δ_(min)between two information symbols S_(i), S_(j) among said informationsymbols greater than a selection threshold, such that:Δ_(min)=min Δ(S _(i) ,S _(j)) with:Δ(S_(i),S_(j))=D(S_(i),S_(j))+D(T(S_(i)),T(S_(j))) D(S_(i),S_(j)) beinga distance between said information symbols S_(i), S_(j) beforetransformation, D(T(S_(i)),T(S_(j))) being a distance between saidinformation symbols S_(i), S_(j) after transformation by the function T.8. The method according to claim 7, wherein the dispersion between twoinformation symbols S_(i), S_(j) is expressed in the form:Δ(S _(i) ,S _(j))=D(S _(i) ,S _(j))+D(T(S _(i)),T(S _(j))) with:${D\left( {S_{i},S_{j}} \right)} = {{\sum\limits_{k = 1}^{n}{d_{euc}^{2}\left( {S_{i,k}^{m},S_{j,k}^{m}} \right)}} = {\sum\limits_{k = 1}^{n}\left\lbrack {\left( {I_{S_{i,k}^{m}} - I_{S_{j,k}^{m}}} \right)^{2} + \left( {Q_{S_{i,k}^{m}} - Q_{S_{j,k}^{m}}} \right)^{2}} \right\rbrack}}$$\begin{matrix}{{D\left( {{T\left( S_{i} \right)},{T\left( S_{j} \right)}} \right)} = {\sum\limits_{k = 1}^{n}{d_{euc}^{2}\left( {{T\left( S_{i} \right)}_{k}^{m},{T\left( S_{j} \right)}_{k}^{m}} \right)}}} \\{= {\sum\limits_{k = 1}^{n}\left\lbrack {\left( {I_{{T(S_{i})}_{k}^{m}} - I_{{T(S_{j})}_{k}^{m}}} \right)^{2} + \left( {Q_{{T(S_{i})}_{k}^{m}} - Q_{{T(S_{j})}_{k}^{m}}} \right)^{2}} \right\rbrack}}\end{matrix}$ S^(m) _(i,k) respectively S^(m) _(j,k), for k ranging from1 to n, the n modulation symbols associated with the information symbolS_(i), respectively S_(j), before transformation, T(S_(i))^(m) _(k),respectively T(S_(j))^(m) _(k), for k ranging from 1 to n, the nmodulation symbols associated with the information symbol S_(i),respectively S_(j), after transformation, I_(x) and Q_(x) are thein-phase and quadrature components of a signal x in the consideredconstellation.
 9. The method according to claim 8, wherein saidtransformation minimises the number of zero terms in the expression ofthe dispersion Δ(S_(i),S_(j)).
 10. The method according to claim 8,wherein the method comprises selecting said transformation from severalavailable transformations, said selection maximising the value of thenon-zero terms in the expression of the dispersion Δ(S_(i),S_(j)).